Selection for Multivariate Copula Based on Conditional Probability Integral Transformation
Bibliographic record
Abstract
The dependence structure among multivariate financial assets is a critical factor for achieving accuracy in the integrated risk measurement.Copula function is a very useful tool to describe the dependence structure between risk factors and plays an important role in the field of financial risk management.When using the Copula model,the most important thing is to judge which Copula is more suitable to describe the data dependence structure.Therefore,it is very important to do research on the selection criteria of multivariate Copula models and the goodness-of-fit test methods.However,in the field of financial risk management,most of the research has focused on bivariate cases.There is very little research on the goodness-of-fit and empirical analysis on the multivariate cases.There is still no effective solutions for the selection and goodness-of-fit test of multivariate Copula functions. Therefore,this paper proposes a selection criterion for Copula's goodness-of-fit based on the method of conditional probability integral transformation.We analyze and compare the Anderson-Darling(AD),Kolmogorov-Smirnov(KS) and Cramer-von Mises(CM) test statistics under the CPIT method with various sample sizes and different variable dimensions.In addition,we use daily data of three stock indices: SP/TSX Composite index(GSPTSE) in Canada's stock market,INMEX.MX in Mexico's stock market and NASDAQ-100(NDX) in America's stock market.Our samples consist of 1606 adjusted-closing price in each of the three indices.We compare the CPIT test statistics with two other methods based on kernel the density estimate and the maximum likelihood estimate. The empirical studies results show that in terms of the power of goodness-of-fit test,the approach we proposed has a better performance.This method is able to solve the puzzle in selecting multivariate Copula models and its goodness-of-fit test is accurate and stable.Specifically,CM test statistic is more powerful in small samples;however in large samples the test is weaker than AD and KS tests.We also show that the AD test has a strong testing ability in large samples.On the other hand,the statistics based on the kernel density estimate method is more appropriate for selecting the best Copula functions under a large sample.The reason is that in a large sample,the kernel function selection has little effect on the estimated distribution.However,in a small sample,the choice of bandwidth in kernel estimation has a big effect on the estimation of marginal distribution,which may result in an unstable result.Although the test based on maximum likelihood estimate is able to choose Gauss Copula function as the best Copula to describe the correlation pattern between datasets,the method is quite unstable.As a result,its capacity of choosing the optimal Copula function is relatively weak.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.035 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".